WARNING: The following contains spoilers for Evangelion: 3.0+1.01 Thrice Upon a Time, now streaming on Amazon Prime Video.
The title Neon Genesis Evangelion roughly translates in Latin to "First Book of the New Gospel." What exactly that meant in the context of the series, as with so many other aspects of Evangelion, has long been up for debate. Some look at the series as a retelling and deconstruction of the Adam and Eve story, while others dismiss the anime's religious symbolism as being purely aesthetic. Evangelion: 3.0+1.01 Thrice Upon a Time, the definitive ending to the franchise, finally gives a direct definition of what a "Neon Genesis" is in the context of Evangelion.
The answer is almost deceptively simple: the "Neon Genesis" is a new creation of the universe. SEELE and NERV's Human Instrumentality Project had the world essentially caught in a loop, repeating the story of Evangelion multiple times. This time around, however, Shinji chooses to restart the world by erasing the Evangelions from existence, freeing himself and his loved ones from the suffering of the time loop they've been caught in. Shinji announces his plan to Rei as the world around them falls apart into a live-action movie set (the same one in which director Hideaki Anno shot the movie's motion capture scenes) and scenes from the previous Evangelion anime and manga are projected on the walls. Rei fittingly describes the plan as "Birth of a new world. Neon Genesis."
This version of Rei reverts to Yui, Shinji's mother from whom all the Reis were originally cloned. Yui reunites with Gendo, fulfilling Gendo's ultimate goal throughout the entire series, and together, both parents sacrifice themselves and their EVA Unit to save all life on Earth. Shinji returns to the beach from The End of Evangelion, which slowly fades into rough animation and storyboards before Mari arrives and restores color to Shinji's perception of the world. Together, the two former pilots say their happy goodbyes to all the Evangelion units. In the film's final scene, Shinji and Mari meet each other at a train station, fully grown, ready to explore this beautiful new world (represented by live-action footage of Anno's actual hometown) together.
There are parallels between this ending and Evangelion's previous animated endings, Episode 26 of the TV series and The End of Evangelion, but the sense of rebirth in those endings was limited by virtue of Shinji's perspective. The TV finale introduces the idea of multiple universes, but due to the abstract presentation, it's nearly impossible to tell if Shinji has actually remade the world beyond himself in any meaningful way or if he's merely adopted a more positive perspective of the world he's living in. The End of Evangelion discusses how "The fate of destruction is also the joy of rebirth," but the film's dramatic emphasis is firmly on the side of destruction rather than creation, and even Shinji's rejection of Instrumentality only manages to save Asuka.
In Evangelion: 3.0+1.01 Thrice Upon a Time, Shinji escapes the trap of only focusing on himself and is now able to save everyone else, not only from one apocalypse but from multiple ones. Neon Genesis Evangelion has always been a very self-involved story for both Shinji and Hideaki Anno, but now both the protagonist and the director have grown beyond the need to stay stuck in the same story. The "Neon Genesis" is the ultimate end of all things Evangelion, and it's the happiest ending to the legendary anime series that one could ever possibly hope for.
All four Rebuild of Evangelion films are streaming worldwide on Amazon Prime Video.